Let $p \in \lbrace2,3,4,...\rbrace$. Suppose that for all $x,y \in \mathbb{Z}$, if $p \mid xy$, then $p \mid x \vee p \mid y$. Show that $p$ is prime.

score 370 · Answer 9 · 2026-04-03 00:54:06

370

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Prove that as $x\to\infty $, $\sum\limits_{p \leq x} \frac{1}{p \log \log p} \approx \log \log \log x$

Published on 03 Apr 2026 - 0:54

score 276 · Answer 10 · 2026-03-26 11:06:49

276

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How probable is that a randomly typed 47 digit odd integer is a prime?

Published on 26 Mar 2026 - 11:06

score 872 · Answer 11 · 2026-03-27 09:48:06

872

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A little more on $\sqrt[3]{\cos\bigl(\tfrac{2\pi}7\bigr)}+\sqrt[3]{\cos\bigl(\tfrac{4\pi}7\bigr)}+\sqrt[3]{\cos\bigl(\tfrac{8\pi}7\bigr)}$

Published on 27 Mar 2026 - 9:48

score 563 · Answer 12 · 2026-03-26 09:42:16

563

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What's so special about primes $x^2+27y^2 = 31,43, 109, 157,\dots$ for cubics?

Published on 26 Mar 2026 - 9:42

score 1k · Answer 13 · 2026-04-11 02:05:52

1k

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Choosing primes uniformly at random

Published on 11 Apr 2026 - 2:05

score 126 · Answer 14 · 2014-12-16 13:49:23

126

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Proving $m$ is prime when $a^{m-1}\equiv 1\pmod m$ and factors of $m-1$ satisy $a^n\equiv r\pmod m,r\neq1$

Published on 16 Dec 2014 - 13:49

score 534 · Answer 15 · 2026-04-09 13:34:12

534

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Extending the zeta function to semiprimes, etc.

Published on 09 Apr 2026 - 13:34

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Let $p \in \lbrace2,3,4,...\rbrace$. Suppose that for all $x,y \in \mathbb{Z}$, if $p \mid xy$, then $p \mid x \vee p \mid y$. Show that $p$ is prime.

Prove that as $x\to\infty $, $\sum\limits_{p \leq x} \frac{1}{p \log \log p} \approx \log \log \log x$

How probable is that a randomly typed 47 digit odd integer is a prime?

A little more on $\sqrt[3]{\cos\bigl(\tfrac{2\pi}7\bigr)}+\sqrt[3]{\cos\bigl(\tfrac{4\pi}7\bigr)}+\sqrt[3]{\cos\bigl(\tfrac{8\pi}7\bigr)}$

What's so special about primes $x^2+27y^2 = 31,43, 109, 157,\dots$ for cubics?

Choosing primes uniformly at random

Proving $m$ is prime when $a^{m-1}\equiv 1\pmod m$ and factors of $m-1$ satisy $a^n\equiv r\pmod m,r\neq1$

Extending the zeta function to semiprimes, etc.

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